Tree stumps can last for years if they are cut high and left untreated. In some cases, of course, they can be handy or at least harmless, depending on your point of view. But if you want camouflage, plant Virginia creeper at the stump base and let it grow up and over. Eventually it will root through the wood , destroying it.
Meanwhile, the creeper will be very attractive, especially in the fall with its deep-red coloring. Too, you can still use the stump for a bench or table.
There are several ways to remove a stump, but I would recommend explosives only in the hands of experts and well away from people or buildings. Even a tractor and chain or other mechanical equipment sounds drastic.
If you can cut the stump back to ground level and cover it with soil, it will rot away in a year or two, especially if you keep the soil moist. Drilling vertical holes through the surface will hurry the process along.
A good way to burn out a stump is to drill similar holes, fill them with potassium (also called saltpeter and usually available from a drugstore), and close the top with clay. After after three months, douse the top of the stump with charcoal lighter and burn.
You also can take an old 5-gallon metal bucket, remove the top and bottom, and punch draft holes in the side. Then place this ''stove'' on top of the stump and build a fire inside. Move the stove as necessary. If your stump is close to the kitchen, you could do your outdoor cooking there at the same time. If you dump your charcoal fire after cooking on an outside grill, you might as well dump it onto the stump while the coals are still burning.
Chemicals are also available for disintegrating stumps. Read the label before you buy one and then proceed accordingly.
Tree roots in your sewer lines can be a more immediate and expensive problem. A good preventive measure is to pour one pound of quarter-inch crystals of copper sulfate (also available at a drugstore) into the toilet bowl twice a year. Never use it in a sink or on metal. Agitate with a plunger before flushing and be sure all the crystals are flushed away so they will not stain or damage the porcelain.